CHAPTER 13
PROBABILISTIC RISK ANALYSIS
The objectives are to:
 Use of statistical and probability concepts in decision
making involving risk and uncertainty
 Illustrate how they can be applied in engineering economy
analysis
 Discuss the limitations relative to their application
RANDOM VARIABLES
Factors having probabilistic outcomes
 The probability that a cost, revenue, useful life, or other
economic factor value will occur, is usually considered to be the
estimated likelihood that an event (value) occurs
 The information about these random variable that is particularly
helpful in decision making are the expected values and
variances
 These values for random variables are used to make the
uncertainty associated with each alternative more explicit
RANDOM VARIABLES
 Capital letters such as X, Y, and Z are used to represent
random variables
 Lower-case letters (x,y,z) denote the particular values that
these variables take on in the sample space (I.e., the set of
possible outcomes for each variable
 When random variable X follows some discrete probability
distribution, its mass function is usually indicated by p(x)
and its cumulative distribution function by P(x)
 When X follows a continuous probability distribution, its
probability density function and its cumulative distribution
function are usually indicated by f(x) and F(x), respectively
DISCRETE RANDOM VARIABLES
 A random variable X is discrete if it can take on a finite number
of values (x1,x2…xL)
 The probability that a discrete random variable X takes on the
value xi is given by
Pr{X = xi} = p(xi) for i = 1,2,….,L (i is a sequential index of the
discrete values, xi, that the variable takes on)
where p(xi) > 0 and i p(xi) = 1
CONTINUOUS RANDOM VARIABLES
A random variable is continuous if:
c
 Pr{c < X < d} = f(x)dx
d
 In the nonnegative function f(x),this is the probability that X is within the set of
real numbers (c,d)

  f(x)dx = 1
-
 The probability that the value X is less than or equal x = k, the cumulative
distribution function F(x) for a continuous case is
d
 Pr{X < k} = F(k) =  f(x)dx
d
-
 Pr{c < X < d} = f(x)dx = F(d) – F( c )
c
In most applications, continuous random variables represent measured data,
such as time, cost and revenue on a continuous scale
MATHEMATICAL EXPECTATIONS AND SELECTED
STATISTICAL MOMENTS
 The expected value of a single random variable X, (E(X), is a
weighted average of the distributed values x that it takes on and
is a measure of the central location of the distribution
 E(X) is the first moment of the random variable about the origin
and is called the mean of the distribution
 E(X) = i xi p( xi ) for x discrete and i = 1,2,…,L

 E(X) =  xf(x)dx for x continuous
-
MATHEMATICAL EXPECTATIONS AND SELECTED
STATISTICAL MOMENTS
 From binomial expansion of [X – E(X)]2, it can be easily shown
that:
 V(X) = E(X2) – [E(X)]2
 V(X) is the second moment of the random variable around the
origin : the expected value of X2, minus the square of its mean
 V(X) is the variance of the random variable X
 V(X) = i x2p(xi) – [E(X)]2 for x discrete

 V(X) =  xi2f(x)dx – [E(X)]2 for x continuous

 The standard deviation of a random variable, SD(X) is the
positive square root of the variance
 SD(X) = [V(X)]1/2
MULTIPLICATION OF A RANDOM VARIABLE BY A
CONSTANT
When a random variable, X, is multiplied by a constant, c, the
expected value E(cX), and the variance, V(cX) are:
 E(cX) = cE(X) = cxi p(xi) for discrete
 E(cX) = cE(X) = cx f(x)dx for continuous
 V(cX) = E{ [cX – E(cX)]2 }
=c2E{ [X – E(X)]2 }
=c2 V (X)
MULTIPLICATION OF TWO INDEPENDENT VARIABLES
When a random variable, Z, is a product of two independent
random variables, X and Y, the expected value, E(Z), and the
variance, V(Z) are
 Z= XY
 E(Z) = E(X) E(Y)
 V(Z) = V(X) [E(Y)]2 + V(Y) [E(X)]2 + V(X) V(Y)
EVALUATION OF PROJECTS WITH DISCRETE RANDOM
VARIABLES
 Expected value and variance concepts apply theoretically to
long-run conditions in which it is assumed that the event is going
to occur repeatedly
 However, application of these concepts is often useful when
investments are not going to be made repeatedly over the long
run
Problem 13-4
In a building project, the amount of concrete to be poured during the next
week is uncertain. The foreman has estimated the following probabilities:
Amount (cubic yards)
1000
1200
1300
1500
2000
probabilities
0.1
0.3
0.3
0.2
0.1
Determine the expected value (amount) of concrete to be poured next
week. Also compute the variance and standard deviation of the amount
of concrete to be poured.
Solution:
Let X = amount of concrete to be poured
E(X) = 1000 (0.1) + 1200 (0.3) + … +2000 (0.1)
= 1,350 cubic yards
V(X) = [(1000)2 (0.1) + (1200)2(0.3) +…+(2000)2 (0.1)] – (1350)2
= 66500
SD(X) = (66500)1/2 = 258
EVALUATION OF PROJECTS WITH CONTINUOUS RANDOM
VARIABLES
Two Frequently Used Assumptions
 Uncertain cash-flow amounts are distributed according to the
normal distribution
 Uncertain cash flow amounts are statistically independent
 No correlation between cash flow amounts is assumed
EVALUATION OF PROJECTS WITH CONTINUOUS RANDOM
VARIABLES
 If there is a linear combination of two or more independent cash
flow amounts (i.e., PW = c0F0 + … +cNFN, where ck values are
coefficients and Fk values are periodic net cash flows) the
expression V(PW) reduces to
N
 V(PW) =
k=0
N
 E(PW) =
ck2 V(Fk)
k=0 ckE(Fk)
Problem 13-6
A small dam is being planned for a river tributary that is subject to
frequent flooding. From past experience, the probabilities that water flow
will exceed the design capacity of the dam during a year, plus relevant
cost information, are as follows:
Design
A
B
C
D
E
Probability of greater
flow during a year
0.10
0.05
0.025
0.015
0.006
Capital
investment
-$180,000
- 195,000
- 208,000
- 214,000
- 224,000
Estimated annual damages that occur if water flows exceed design
capacity are $150,000, 160,000, 170,000, 190,000, and 210,000 for
design A, B, C, D, and E, respectively. The life of the dam is expected to
be 50 years, with negligible salvage value. For an interest rate of 8% per
year, determine which design should be implemented. What nonmonetary considerations might be important to the selection?
Solution:
Prob. of
Annual
Total
Greater
Capital
Estimated
Expected
Equivalent
Flow during Investment Damage
Cost of
Present
Design a year
Damages
Worth
----------------------------------------------------------------------------------------------------A 0.1
-180,000
-150,000
-15,000
-363,000
B 0.05
-195,000
-160,000
-8,000
-292,000
C 0.025
-208,000
-175,000
-4,375
-261,522
D 0.015
-214,000
-190,000
-2,850
-248,865
E 0.006
-224,000
-210,000
-1,260
-239,414
EVALUATION OF UNCERTAINTY USING MONTE CARLO
SIMULATION
 Computer-assisted simulation tool for analyzing more complex
project uncertainties
 Monte Carlo simulation generates random outcomes for
probabilistic factors which imitate the randomness inherent in
the original problem
MONTE CARLO SIMULATION
 Can be used only when a process has a random, or chance,
component
 Based on a probabilistic distribution
 Random samples taken from this probability distribution are
analogous to observations made on the system itself
 As the number of observations increases, the results of the
simulation will more closely approximate the random behavior of
the real system
BASIC STEPS IN MONTE CARLO SIMULATION
 Identify a probability distribution for each random component of
the system
 Work out an assignment so that intervals of random numbers
will correspond to the probability distribution
 Obtain the random numbers needed for the study
 Interpret the results
SOURCE OF RANDOM NUMBERS
 Large studies
Computer-generated random numbers
 Small studies
Table of random digits
Example
The manager of a machine shop is concerned about machine
breakdowns. A decision has been made to simulate to
breakdowns for a 1-day period. Historical data on breakdowns
over the last 100 days are given in the following table:
Number of Breakdowns
0
1
2
3
4
5
frequency
10
30
25
20
10
5
Simulate breakdowns for a 10-day period.
Solution
Number of
Breakdowns Frequency
0
10
1
30
2
25
3
20
4
10
5
5
Probability
.10
.30
.25
.20
.10
.05
Cumulative
Probability
.10
.40
.65
.85
.95
1.00
Using the table of random digits
18 25 73 12 54 96 23 31 45 01
Corresponding
Random
numbers
00 to 09
10 to 39
40 to 64
65 to 84
85 to 94
95 to 99
Day
Random Number
Stimulated number of breakdowns
-------------------------------------------------------------------------------------1
2
3
4
5
6
7
8
9
10
18
25
73
12
54
96
23
31
45
01
1
1
3
1
2
5
1
1
2
0
Mean number of breakdowns = (17/10) = 1.7
Expected number of breakdowns =
0*.1 + 1*.3+ 2*.25 +3*.2+ 4*.1+5*.05=2.05 per day
SIMULATING THEORATICAL DISTRIBUTIONS
 Poisson distribution
 Exponential distribution
 Normal distribution
Use a table of normally distributed random numbers
See Table 13-10
Simulated value = mean + Random number x standard deviation
 Uniform distribution
Simulated value = A + (RN/RNm )[B-A]
where
RNm = maximum possible random number
RN = selected random number
A = minimum outcome
B = maximum outcome
USING MONTE CARLO SIMULATION IN ENGINEERING
PROJECTS
 Construct an analytical model that represents the actual decision
situation
 Develop a probability distribution from subjective or historical
data for each uncertain factor in the model
 Sample outcomes are randomly generated by using probability
distribution for each uncertain quantity and then used to
determine a trial outcome for the model
 Repeating sampling process many times leads to a frequency
distribution of trial outcomes, which are used to make
probabilistic statements
EXAMPLE 13-9
The following estimates relate to an engineering project being considered by a
large manufacturer of air-conditioning equipment. Subjective probability functions
have been estimated for the four independent uncertain factors as follows:
CAPITAL INVESTMENT Normally distributed with a mean of –$50,000 and a
standard deviation of $1,000.
USEFUL LIFE Uniformly and continuously distributed with a minimum life of 10
years and maximum life of 14 years
ANNUAL REVENUE
$35,000 with a probability of 0.4
$40,000 with a probability of 0.5
$45,000 with a probability of 0.1
ANNUAL EXPENSES Normally distributed, with a mean of -$30,000 and a
standard deviation of $2000.
The management of the company wishes to determine whether or not the capital
investment in the project will be a profitable one. MARR = 10%.
Solution
Five trial outcomes are computed manually
Random
capital
Project
project
Normal
investment
life, N
life, N
Trail
deviate
[-50,000+
[10+(RN/999)
(nearest
Number (RND1)
RND1(10,000)
3-digit RNs (14-10)]
integer)
-----------------------------------------------------------------------------------------------------------1 +1.003
-$48,997
807
13.23
13
2 +0.358
- 49,642
657
12.63
13
3 -1.294
- 51,294
488
11.95
12
4 +0.019
- 49,981
282
11.13
11
5 -0.147
- 50,147
504
12.02
12
------------------------------------------------------------------------------------------------------------
Annual revenue
-$35,000 for 0-3
Annual Expense
1-digit
-40,000 for 4-5
[-30,000 +
PW
RN
45,000 for 9
RND2 RND2(20,000)
(R+E)(P/A,10,N)
----------------------------------------------------------------------------------------------------------1
2
$35,000
+0.036 -$29,928
-$12,969
2
0
35,000
+0.605 - 31,000
- 22,720
3
4
40,000
-1.470 - 32,940
- 3,189
4
9
45,000
-1.864 - 33,728
+ 23,232
5
8
40,000
+1.223 - 27,554
+ 34, 656
-----------------------------------------------------------------------------------------------------------Total
+19,010
Estimated average present worth = (19,010/5) = $3,802
DECISION TREES
 Also called decision flow networks and decision diagrams
 Powerful means of depicting and facilitating analysis of
important problems, especially those that involve sequential
decisions and variable outcomes over time
 Practical tool because it permits large complicated problems to
be reduced to a series of smaller simple problems
 Enable objective analysis and decision making that includes
explicit consideration of the risk and effect of the future
GENERAL PRINCIPLE OF DIAGRAMING
 The Decision Tree Diagram Should Show the Following
 (With square symbol to depict decision node and circle symbol
to depict chance outcome node):
 All initial or immediate alternatives among which the decision
maker wishes to choose
 All uncertain outcomes and future alternatives the decision
maker wishes to consider
 Note alternatives at any point and outcomes at any chance
outcome node must be:
 Mutually exclusive
 Collectively exhaustive; that is, one event must be chosen or
something must occur if the decision point or outcome node is
reached